Rotating Bumblebee Black Hole

• Rotating Bumblebee Black Hole is a spacetime solution in bumblebee gravity where a vector field attains a nonzero vacuum value that spontaneously breaks Lorentz symmetry.
• The models span 3D BTZ-like, 4D Kerr-like, and slowly rotating metrics, each modifying horizon structures, ergospheres, and thermodynamic properties.
• Perturbative analyses, quasinormal mode studies, and geodesic investigations reveal that the Lorentz-violating parameter significantly influences decay rates, shadow deformations, and energy extraction efficiencies.

Rotating bumblebee black hole denotes a rotating black-hole spacetime in bumblebee gravity, a vector-tensor theory in which a bumblebee field $B_\mu$ acquires a nonzero vacuum expectation value $\langle B_\mu\rangle=b_\mu$ and spontaneously breaks Lorentz symmetry. In the literature, the Lorentz-breaking strength is written as $l$, $\ell$, or $s=\xi b^2$, and the expression covers several distinct geometries: rotating BTZ-like black holes in three-dimensional Einstein-bumblebee gravity, Kerr-like and slowly rotating black holes in four-dimensional Einstein-bumblebee models, charged and AdS generalizations, Kerr-Sen-like and Kerr-Newman-AdS deformations, metric-affine bumblebee solutions, and five-dimensional equal-angular-momentum black holes (Ding et al., 2023, Wang et al., 2021, Chen et al., 2 Jul 2026).

1. Theoretical framework and Lorentz-symmetry breaking

A representative four-dimensional bumblebee action used in rotating black-hole studies is

$$S=\int d^4 x \sqrt{-g} \left[ \frac{1}{16\pi} \left( R+\varrho B^\mu B^\nu R_{\mu\nu} \right) -\frac{1}{4}B^{\mu\nu}B_{\mu\nu} - V(B^\mu) \right],$$

with

$$B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.$$

The potential is chosen so that the field freezes at a symmetry-breaking minimum, $\langle B^\mu\rangle=b^\mu$, and the Lorentz-violating parameter is then packaged as $\ell=\varrho b^2$ or $l=\xi b^2$ in the notation of individual papers (Gu et al., 2022, Islam et al., 2024).

In the three-dimensional Einstein-bumblebee literature, the same structure appears with a negative cosmological constant and a nonminimal coupling $\langle B_\mu\rangle=b_\mu$0. The exact rotating BTZ-like solution of one branch requires a linear functional potential, whereas the quadratic potential forces $\langle B_\mu\rangle=b_\mu$1 and does not produce an AdS BTZ-like black hole (Ding et al., 2023). In the perturbative BTZ-like branch used for quasinormal-mode analyses, the Lorentz-violation parameter is often denoted by $\langle B_\mu\rangle=b_\mu$2, with $\langle B_\mu\rangle=b_\mu$3, while $\langle B_\mu\rangle=b_\mu$4 denotes the AdS radius (Chen et al., 2023).

The metric-affine formulation introduces a different realization of the same symmetry-breaking mechanism. There the action contains the traceless coupling

$\langle B_\mu\rangle=b_\mu$5

and the connection enjoys projective invariance. The corresponding rotating black hole is Kerr-like but contains additional anisotropic structures, including an off-diagonal $\langle B_\mu\rangle=b_\mu$6 term (Nascimento et al., 30 Mar 2026).

2. Geometric families and representative metrics

The literature does not use a single universal rotating bumblebee metric. Instead, several solution classes are employed, often with different assumptions about dimension, matter sector, cosmological constant, and the status of the field equations.

Sector	Representative metric feature	Distinctive point
3D rotating BTZ-like	$\langle B_\mu\rangle=b_\mu$7	Exact AdS solution with bumblebee coupling (Ding et al., 2023)
4D Kerr-like Einstein-bumblebee	$\langle B_\mu\rangle=b_\mu$8	Widely used in QPO, disk, and reflection studies (Wang et al., 2021)
4D slow rotation	$\langle B_\mu\rangle=b_\mu$9, $l$0	Exact only in slow-rotation regime (Ding et al., 2020)
4D alternative Kerr-like deformations	modified $l$1, scalar-gradient, or metric-affine terms	Used in shadow and precession studies (Islam et al., 2024, Ou et al., 1 Apr 2026, Nascimento et al., 30 Mar 2026)
5D equal angular momenta	$l$2	Kerr/CFT testbed with Komar/Wald mismatch (Chen et al., 2 Jul 2026)

For the exact three-dimensional BTZ-like branch, the metric takes the form

$l$3

and reduces to the standard rotating BTZ black hole as $l$4 (Ding et al., 2023). A different BTZ-like branch used in scalar and higher-spin perturbation analyses keeps the lapse function BTZ-like and modifies only the radial sector through $l$5 (Chen et al., 2023, Quan et al., 26 Mar 2026).

The frequently used four-dimensional Kerr-like Einstein-bumblebee metric is

$l$6

with

$l$7

It is Kerr-like because it reduces to Kerr at $l$8 and preserves stationarity, axisymmetry, and frame dragging (Wang et al., 2021).

A separate line of work emphasizes that an exact fully rotating four-dimensional solution is not established in Einstein-bumblebee theory. The controlled result is the slowly rotating metric

$l$9

obtained by solving the $\ell$0 and $\ell$1 equations. In the purely radial bumblebee configuration this slow-rotation solution exists for arbitrary $\ell$2, while with an angular bumblebee component it exists only if $\ell$3 is as small as or smaller than the rotation parameter $\ell$4 (Ding et al., 2020). The same concern reappears in later phenomenological work, which states that the exact rotating metric does not fully satisfy all field equations and therefore adopts a slowly rotating spacetime instead (Mangut et al., 2023).

3. Horizons, ergospheres, thermodynamics, and holography

In the four-dimensional Kerr-like Einstein-bumblebee metric, the horizons are determined by $\ell$5,

$\ell$6

and the spin bound becomes

$\ell$7

Negative $\ell$8 enlarges the allowed spin range, while positive $\ell$9 restricts it (Wang et al., 2021). In the modified-mass-function rotating black hole in bumblebee gravity (RBHBG), the event horizon is

$s=\xi b^2$0

and the extremal case with $s=\xi b^2$1 can have $s=\xi b^2$2 (Islam et al., 2024).

Three-dimensional BTZ-like constructions display two distinct patterns. In the exact branch of the rotating BTZ-like solution, the horizons and ergosphere depend on the bumblebee coupling constant $s=\xi b^2$3, and the ergosphere radius is written as $s=\xi b^2$4 (Ding et al., 2023). In the BTZ-like branch used for scalar and higher-spin quasinormal modes, the horizon radii

$s=\xi b^2$5

or equivalently

$s=\xi b^2$6

are independent of $s=\xi b^2$7 or $s=\xi b^2$8, while the Lorentz-violating imprint sits in the radial metric coefficient; these spacetimes are singular at $s=\xi b^2$9 or $$S=\int d^4 x \sqrt{-g} \left[ \frac{1}{16\pi} \left( R+\varrho B^\mu B^\nu R_{\mu\nu} \right) -\frac{1}{4}B^{\mu\nu}B_{\mu\nu} - V(B^\mu) \right],$$0, so the analyses impose $$S=\int d^4 x \sqrt{-g} \left[ \frac{1}{16\pi} \left( R+\varrho B^\mu B^\nu R_{\mu\nu} \right) -\frac{1}{4}B^{\mu\nu}B_{\mu\nu} - V(B^\mu) \right],$$1 or $$S=\int d^4 x \sqrt{-g} \left[ \frac{1}{16\pi} \left( R+\varrho B^\mu B^\nu R_{\mu\nu} \right) -\frac{1}{4}B^{\mu\nu}B_{\mu\nu} - V(B^\mu) \right],$$2 (Chen et al., 2023, Quan et al., 26 Mar 2026).

Thermodynamics is likewise model-dependent. For the exact rotating BTZ-like black hole, the horizon “area” and volume must be redefined because of the nontrivial coupling between the bumblebee field and the Ricci tensor: $$S=\int d^4 x \sqrt{-g} \left[ \frac{1}{16\pi} \left( R+\varrho B^\mu B^\nu R_{\mu\nu} \right) -\frac{1}{4}B^{\mu\nu}B_{\mu\nu} - V(B^\mu) \right],$$3 With these definitions, the entropy-area relation, first law, and Smarr formula are restored, and the entropy product of the inner and outer horizons is universal. The dual CFT then has equal central charges, $$S=\int d^4 x \sqrt{-g} \left[ \frac{1}{16\pi} \left( R+\varrho B^\mu B^\nu R_{\mu\nu} \right) -\frac{1}{4}B^{\mu\nu}B_{\mu\nu} - V(B^\mu) \right],$$4 (Ding et al., 2023).

In five-dimensional Einstein-bumblebee gravity with equal angular momenta, the macroscopic thermodynamics depends on the chosen charge prescription. The Wald and Komar definitions differ by a constant prefactor determined solely by the bumblebee coupling, with

$$S=\int d^4 x \sqrt{-g} \left[ \frac{1}{16\pi} \left( R+\varrho B^\mu B^\nu R_{\mu\nu} \right) -\frac{1}{4}B^{\mu\nu}B_{\mu\nu} - V(B^\mu) \right],$$5

The Kerr/CFT calculation reproduces the Komar entropy rather than the Wald entropy, so the microscopic entropy matches the area-law version (Chen et al., 2 Jul 2026).

Charged AdS extensions add a further layer. In the Bumblebee Kerr-Newman-AdS black hole, the Lorentz-violating parameter modifies the horizon polynomial, shifts extremality, changes the Hawking temperature profile, affects the entropy

$$S=\int d^4 x \sqrt{-g} \left[ \frac{1}{16\pi} \left( R+\varrho B^\mu B^\nu R_{\mu\nu} \right) -\frac{1}{4}B^{\mu\nu}B_{\mu\nu} - V(B^\mu) \right],$$6

and is reported to induce black hole remnants and extended stability phases (Hassanabadi et al., 20 Dec 2025).

4. Perturbations, quasinormal modes, superradiance, and scalar clouds

The most systematic perturbative results are available for rotating BTZ-like black holes. For scalar perturbations, the exact quasinormal frequencies show that the Lorentz-symmetry-breaking parameter imprints only in the imaginary parts; the real parts remain those of the usual rotating BTZ black hole. Negative $$S=\int d^4 x \sqrt{-g} \left[ \frac{1}{16\pi} \left( R+\varrho B^\mu B^\nu R_{\mu\nu} \right) -\frac{1}{4}B^{\mu\nu}B_{\mu\nu} - V(B^\mu) \right],$$7 makes the perturbation decay more rapidly, whereas positive $$S=\int d^4 x \sqrt{-g} \left[ \frac{1}{16\pi} \left( R+\varrho B^\mu B^\nu R_{\mu\nu} \right) -\frac{1}{4}B^{\mu\nu}B_{\mu\nu} - V(B^\mu) \right],$$8 makes it decay more slowly. The same analysis rewrites the frequencies in left-moving and right-moving CFT form and finds that the Lorentz-symmetry-breaking parameter enhances the conformal weights $$S=\int d^4 x \sqrt{-g} \left[ \frac{1}{16\pi} \left( R+\varrho B^\mu B^\nu R_{\mu\nu} \right) -\frac{1}{4}B^{\mu\nu}B_{\mu\nu} - V(B^\mu) \right],$$9 and $$B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.$$0 (Chen et al., 2023).

This picture was extended to massive scalar, fermionic, and vector perturbations. In that broader treatment, the real parts of the quasinormal frequencies depend only on the angular quantum number and are the same as those in the standard BTZ black hole, while the imaginary parts carry the Lorentz-violating dependence. A special exception occurs for vector perturbations: for the fundamental modes $$B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.$$1, the left-moving frequencies with positive mass and the right-moving ones with negative mass have imaginary parts independent of $$B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.$$2. The same work also states that the universal relation $$B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.$$3 survives for scalar, fermionic, and vector operators in the dual CFT (Quan et al., 26 Mar 2026).

Stationary scalar clouds around the rotating BTZ-like black hole provide the threshold counterpart of this quasinormal-mode analysis. With Robin boundary conditions at the AdS boundary, only nodeless clouds are found, namely the fundamental overtone $$B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.$$4. The Lorentz-breaking parameter $$B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.$$5 and the azimuthal quantum number $$B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.$$6 play opposite roles in the cloud existence lines, which leads to degenerate scalar clouds: different pairs $$B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.$$7 can yield the same existence line while retaining different radial profiles. In this setup, the superradiance condition itself is unchanged,

$$B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.$$8

but superradiant instabilities appear only for the fundamental modes (Quan et al., 27 Jan 2025).

Four-dimensional slow-rotation studies show an analogous dominance of the damping sector. For scalar and vector perturbations of the slowly rotating Einstein-bumblebee black hole, the Lorentz-violating parameter has a significant effect on the imaginary part of the quasinormal frequencies and a relatively smaller impact on the real part. For axial gravitational perturbations, increasing $$B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.$$9 acts similarly to increasing the rotation parameter $\langle B^\mu\rangle=b^\mu$0. These frequencies were computed numerically with both the matrix method and the continued fraction method (Liu et al., 2022).

5. Geodesics, precession, shadows, disks, and energy extraction

Timelike and null geodesics are among the main probes of rotating bumblebee metrics. In the relativistic precession model applied to the four-dimensional Kerr-like Einstein-bumblebee black hole, the azimuthal frequency decreases with the Lorentz-symmetry-breaking parameter, while both the periastron and nodal precession frequencies increase in the rotating case. In the non-rotating limit, the nodal precession frequency disappears for arbitrary Lorentz-symmetry-breaking parameter (Wang et al., 2021). A different scalar-gradient rotating bumblebee metric gives a complementary pattern: Lorentz violation suppresses the Lense-Thirring precession near the horizon, enhances geodetic precession in the static limit, and increases the periastron precession frequency of bound equatorial orbits (Ou et al., 1 Apr 2026).

Accretion observables also respond directly to the deformation parameter. Thin-disk calculations around the Kerr-like Einstein-bumblebee black hole show that the flux, emission spectrum, and efficiency depend on $\langle B^\mu\rangle=b^\mu$1; for prograde motion the efficiency increases with $\langle B^\mu\rangle=b^\mu$2, whereas for retrograde motion it decreases (Ding et al., 2019). In X-ray reflection spectroscopy, a full bumblebee reflection model implemented in $\langle B^\mu\rangle=b^\mu$3 fits the NuSTAR spectrum of EXO 1846–031 well, but the Lorentz-violating parameter cannot be constrained because of a very strong degeneracy with the spin parameter $\langle B^\mu\rangle=b^\mu$4 (Gu et al., 2022).

Shadow phenomenology is notably model-dependent. In one Kerr-like RBHBG family, increasing $\langle B^\mu\rangle=b^\mu$5 enlarges the shadow radius irrespective of spin or inclination angle and generally increases the distortion while decreasing the event-horizon area (Islam et al., 2024). In the slowly rotating charged Kerr-Newman-like and Kerr-Newman-(A)dS-like bumblebee solutions, the radius of the shadow reference circle decreases with the Lorentz-violating parameter and the charge parameter, while the distortion parameter increases with both (Liu et al., 2024). In the improved Newman–Janis construction applied to the bumblebee model, rotation effects on the shadow are enhanced rather than weakened, and the distortion can rise to about $\langle B^\mu\rangle=b^\mu$6 for larger spin and $\langle B^\mu\rangle=b^\mu$7 or $\langle B^\mu\rangle=b^\mu$8 in the scanned examples (Alexeyev et al., 2024). In the Kerr-Sen-like bumblebee spacetime, positive $\langle B^\mu\rangle=b^\mu$9 shifts the shadow boundary to the right, negative $\ell=\varrho b^2$0 shifts it to the left, and larger $\ell=\varrho b^2$1 increases the deformation while lowering the emission rate and moving its peak toward lower $\ell=\varrho b^2$2 (Jha et al., 2020). In the metric-affine rotating solution, increasing $\ell=\varrho b^2$3 produces vertical flattening, teardrop-shaped deformation, and local collapse of the lower silhouette region (Nascimento et al., 30 Mar 2026). In the scalar-gradient rotating metric, by contrast, the critical curve is nearly unaffected, while the inner shadow shrinks and the lensed ring becomes broader and brighter as $\ell=\varrho b^2$4 increases (Ou et al., 1 Apr 2026).

Energy extraction is equally sensitive to the chosen rotating bumblebee geometry. In the Bumblebee Kerr-Newman-AdS black hole, increasing $\ell=\varrho b^2$5, $\ell=\varrho b^2$6, and $\ell=\varrho b^2$7 enlarges and distorts the ergoregion, intensifies frame dragging, increases the horizon angular velocity

$\ell=\varrho b^2$8

and raises the maximum Penrose efficiency (Hassanabadi et al., 20 Dec 2025). In the Kerr-Sen-like bumblebee spacetime used for magnetic reconnection, larger Lorentz-symmetry breaking and larger Bumblebee charge shrink the horizon, widen the ergoregion, enlarge the allowed extraction region, and increase the covering factor, so extraction becomes more likely to occur closer to the central region (YuChih et al., 27 Oct 2025). This suggests that statements about Penrose or reconnection efficiency are strongly metric-dependent within the broader rotating-bumblebee literature.

6. Observational constraints, phenomenology, and open issues

Current phenomenology uses X-ray timing, X-ray spectroscopy, and horizon-scale imaging to bound the Lorentz-violating parameter. The quasi-periodic-oscillation analysis of GRO J1655-40, XTE J1550-564, and GRS 1915+105 finds that the tightest constraint comes from GRO J1655-40, with a best-fit negative Lorentz-symmetry-breaking parameter. At the same time, $\ell=\varrho b^2$9 remains within the $l=\xi b^2$0 region for all three sources, so general relativity remains consistent with the data (Wang et al., 2021).

Event Horizon Telescope observables have been used in several rotating-bumblebee models. Shadow-radius and distortion methods, as well as shadow-area and oblateness methods, were applied to the RBHBG family to estimate $l=\xi b^2$1 and $l=\xi b^2$2 from M87* and Sgr A* shadow data (Islam et al., 2024). In the rotation-accounting analysis based on an improved Newman–Janis construction, the Sgr A* shadow-size constraints allow all scanned values of $l=\xi b^2$3 at moderate spin, but sufficiently high spin defines a critical $l=\xi b^2$4 and excludes part of parameter space (Alexeyev et al., 2024).

Jet precession provides an additional channel. Modeling the approximately 11-year jet precession period of M87* with spherical orbits in a rotating bumblebee black hole gives warp-radius intervals $l=\xi b^2$5 for prograde motion and $l=\xi b^2$6 for retrograde motion from the jet period alone, and narrower intervals once the EHT shadow angle is included. The analysis states that $l=\xi b^2$7 may indicate a non-vacuum bumblebee vector field near the black hole (Sui et al., 24 Feb 2026).

A central open issue is the status of exact four-dimensional rotating solutions in Einstein-bumblebee gravity. One line of work states that there seems to be no full rotating black hole solution and that the Newman–Janis algorithm cannot be used naively to generate one; on that basis, the slowly rotating solution is treated as the physically consistent result (Ding et al., 2020). Another line of work nonetheless employs Kerr-like metrics generated by modified or improved Newman–Janis procedures, or by alternative geometric ansätze, as phenomenological spacetimes for shadows, precession, and accretion (Islam et al., 2024, Alexeyev et al., 2024). This makes the present literature highly productive but also explicitly model-contingent: observational constraints on a “rotating bumblebee black hole” are constraints on a chosen rotating realization of Lorentz-symmetry breaking rather than on a single universally accepted metric.